2017
Том 69
№ 6

All Issues

Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. I

Bodenchuk V. V., Serdyuk A. S.


Abstract

We prove that the kernels of analytic functions of the form $${H}_{h,\beta }(t)={\displaystyle \sum_{k=1}^{\infty}\frac{1}{ \cosh kh} \cos \left(kt-\frac{\beta \pi }{2}\right),}h>0,\beta \in \mathbb{R},$$ satisfy Kushpel’s condition $C_{y,2n}$ starting from a certain number $n_h$ explicitly expressed via the parameter $h$ of smoothness of the kernel. As a result, for all $n ≥ n_h$ , we establish lower bounds for the Kolmogorov widths $d_{2n}$ in the space $C$ of functional classes that can be represented in the form of convolutions of the kernel $H_{h,β}$ with functions $φ⊥1$ from the unit ball in the space $L_{∞}$.

English version (Springer): Ukrainian Mathematical Journal 67 (2015), no. 6, pp 815-837.

Citation Example: Bodenchuk V. V., Serdyuk A. S. Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. I // Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 719-738.